NDA Practice Questions

The sum of the first \( k \) terms of a sequence \( S \) is \( 3 k^{2 } + 5 k \). Which of the following is correct ?

The sum of the first k terms of a sequence S is 3k² + 5k. Which of the following is correct ?

The sum of the first 8 terms of a GP is five times the sum of its first 4 terms . If \( r \ neq 1 \) is the common ratio, how many possible real values can \( r \ ) have?

The sum of the first 8 terms of a GP is five times the sum of its first 4 terms . If \( r \ neq 1 \) is the common ratio, how many possible real values can \( r \ ) have?

If one root of the equation \( x^{2} - k x + k = 0 \) is \( 2 \sqrt{3 } \) more than the other , then which of the following is the value of \( k \ )?

If one root of the equation \( x^{2} - kx + k = 0 \) is \( 2 \sqrt{3 } \) more than the other , then which of the following is the value of \( k \ )?

If \( x+\frac {5}{y}=4 \ ) and \( y +\frac{5}{ x}=-4 \), then what is the value of \( (x+y) \)?

If \( x+\frac {5}{y}=4 \ ) and \( y +\frac{5}{ x}=-4 \), then what is the value of \( x+y \)?

If the 5th , 7th, and 13th terms of an AP are in GP, what is the ratio of its first term to its common difference ?

If the 5th , 7th, and 13th terms of an AP are in GP, what is the ratio of its first term to its common difference ?

If \( p, 1 , q \) are in AP and \( p, 2, q \) are in GP, then which of the following statements is/are true ?
I. \( p, 4, q \ ) are in HP .
II. \( (1 / p), 1 / 4, (1 / q) \) are in AP.

If \( p, 1 , q \) are in AP and \( p, 2, q \) are in GP, which of the following statements is/are true ?
I. \( p, 4, q \ ) are in HP .
II. \( (1 / p), 1 / 4, (1 / q) \) are in AP.
Use the code given below to choose the correct answer.

If \( x=(111 1)_{2}, y=( 1001)_{2} \ ), and \( z =(110)_{2} \), then what is the value of \( x^{ 3}-y^{3}-z ^{3}-3 x y z \)?

If \(x=(111 1)_{2},\; y =(1001)_{2 }\) and \( z=(110)_{2 }\), then what is the value of \(x^{3 }-y^{3}-z^{ 3}-3xyz\)?

If \[\Delta = \left|\ begin{array }{lll}a & b & c \\d & e & f \\g & h & i\end {array}\right |\]and \(A , B, C, D, G\) are the cofactors of the elements \(a, b, c , d, g\) respectively , then what is the value of \(bB + cC - dD - gG\)?

If \(\Delta =\left|\begin {array}{lll }a & b & c \\ d & e & f \\ g & h & i\end{array }\right|\) and \(A, B , C, D, G\ ) are the cof actors of the elements \( a, b, c, d , g\) respectively , then what is the value of \(b B + c C - d D - g G\)?

Summary
\(\Delta=\ left|\begin {array}{ccc }k(k+2)&2k +1&1\\2k+1 &k+2&1\\3& 3&1\end{array }\right|\)< br>Consider the following statements regarding \(\Delta\ ):
I. \ (\Delta\) is positive if \(k>0\).< br>II. \(\ Delta\) is negative if \(k<0\).< br>III. \(\ Delta\) is zero if \( k=0\).
How many of the above statements are correct ?

Consider the following statements regarding the determinant \(\Delta = \left|\begin {array}{ccc }k(k+2)&2k +1&1\\2k+1 &k+2&1\\3& 3&1\end{array }\right|\):
I. \(\ Delta\) is positive if \(k>0\).< br>II. \(\ Delta\) is negative if \(k<0\).< br>III. \(\ Delta\) is zero if \( k=0\).
How many of the above statements are correct ?

If \[ \left |\begin{array }{ccc} 2 & 3+i & -1\\ 3-i & 0 & i-1\\ -1 & -1-i & 1\\ \end{ array}\right | = A + iB \] where \(i=\sqrt {-1}\), then what is the value of \( A+B\)?

If \[ \left |\begin{array }{ccc} 2 & 3+i & -1 \\ 3-i & 0 & i-1 \\ - 1 & -1-i & 1 \end{array }\right| = A + iB \ ] where \( i = \sqrt {-1} \), then what is the value of \( A + B \)?

If \( A^{2 }+B^{2}+C^{ 2}=0 \), then
\[
\ left|\begin {array}{ccc }
1 & \ cos C & \cos B \\
\ cos C & 1 & \cos A \\
\cos B & \cos A & 1
\end {array}\right |
\]
what is the value of?

If \(A^{2}+ B^{2}+C^{2 }=0\), then what is the value of \ (\displaystyle \left|\begin {array}{ccc }1 & \cos C & \cos B \\ \cos C & 1 & \cos A \\ \cos B & \cos A & 1\end{array }\right|\)?

If \( \omega \) is a non -real cube root of unity , then what is a root of the equation
\[ \left |\begin{array }{ccc} x+1 & \omega & \omega^{2 } \\ \omega & x+\omega ^{2} & 1 \\ \omega^{2 } & 1 & x+\ omega \end {array}\right |=0 \]
< br>?

If \( \omega \) is a non -real cube root of unity , then what is a root of the equation
\[
\ left|\begin {array}{ccc }
x+1 & \omega & \ omega^{2} \ \\
\omega & x+\omega ^{2} & 1 \ \\
\omega ^{2} & 1 & x+\omega
\end{array }\right| = 0
\]

?

What is \( \left(\frac {\sqrt{3}+ i}{\sqrt{3 }-i}\right )^{3} \) equal to?

What is \(\ left(\frac {\sqrt{3}+ i}{\sqrt{3 }-i}\right )^{3}\) equal to?

If \( x^{2 }-x+1=0 \), then
\ [
\left (x-\frac{1 }{x}\right )^{2}+\left (x-\frac{1 }{x}\right )^{4}+\left (x-\frac{1 }{x}\right )^{8}
\ ]
is equal to what?

If \(x^{2}- x+1=0\), then what is the value of \ (\left(x-\ frac{1}{x}\ right)^{2} +\left(x-\ frac{1}{x}\ right)^{4} +\left(x-\ frac{1}{x}\ right)^{8}\ )?